Math Formula

Tuesday, October 2, 2012

Should I Eat That Apple Today Or Tomorrow? - Part II

In last week's post, I discussed the question of a whether a free good should always be consumed, or if it might make sense to abstain from consumption under certain circumstances.

We went through four examples of consumption.
In example 1 and 2, the saturation factor was low (\(s = 0.2\)).
In example 1, the consumer always consumed, and gross utility \(g = 26\).
In example 2, the consumer abstained in period 2, and gross utility \(g = 20\).

In example 3 and 4, the saturation factor was high (\(s = 0.6\)).
In example 3, the consumer always consumed, and gross utility \(g = 18\).
In example 4, the consumer abstained in period 2, and gross utility \(g = 20\).

So, here is the answer to our initial question: You should eat or not eat the apple depending on the saturation factor \(s\) ! For a big \(s\) , not consuming in period 2 was better; for a small \(s\), consuming was better.

However, at which threshold does the decision change?

Let's make a step back first. As you might have noticed, I only varied the decision for period 2, and here's why.

In period 1, consuming is always superior to not consuming. There was no prior period from which the consumer might still be saturated, so he always consumes. Thus, we can assume \(c_1 = 1\), and can disregard the consumption decision from now on.

As we have seen, changing the decision in period 2 has an effect on the gross utility, so \(c_2\) remains a variable to be considered.

However, period 3 can be disregarded again. Why? If \(c_2\) was 0, there is no saturation, and similar to period 1, consumption is better. If \(c_3\) was 1, there is saturation, but there is no 4th period to save consumption for, so again, consumption is better. Thus, \(c_3 = 1\) , and can be disregarded further on.
The only variable to maximize against is \(c_2\).

In the light of the above, let's reconsider our gross utility function:
\(c_1 = 1\)
\(c_2 =\) to be seen
\(c_3 = 1\)
\(u_1 = a c_1  = a\)
\(u_2 = a c_2 (1 - c_1 s) = a c_2 (1 - s)\)
\(u_3 = a c_3 (1 - c_2 s) = a (1 - c_2) s) = a - a c_2 s\)

\(g = u_1 + u_2 + u_3 = a + a _2 (1 - s) +  a - a c_2 s = \)
\(2 a + a c_2 - a c_2 s - a _2 s = \)
\(2 a + a _2 - 2 a _2 s = 2 a + a _2 (1 - 2s)\)

Now we can directly compare the two outcomes with each other; the gross utility in case of consumption in period 2 (which is , and the gross utility in case of no consumption in period 2.

So, for \(c_2 = 1, g\) would be: \(2 a + a (1 - 2s)\)
And for \(c_2 = 0, g\) would be: \(2 a + 0 = 2a\)

So, this question can be formulated as inequation:

\(2 a + a (1 - 2s) > 2 a\)
\(a (1 - 2s) > 0\)
\(1 - 2s > 0\)
\(1 > 2 s\)
\(1/2 > s\)
\(s < 1/2\)

So, if the saturation factor \(s < 1/2\), gross utility is bigger with \(c_2 = 1\).
For \(s > 1/2\), gross utility is bigger with \(c_2 = 0\).
For \(s = 1/2\), the consumer is indifferent, so the gross utility is equal.

Of course, your real saturation factor \(s\) is hardly known. However, I find it quite interesting to keep in mind that for a big saturation factor, I should rather consider not consuming. The bigger the impact on the reduction of the satisfaction of tomorrow's consumption, the more I should be inclined to defer consumption.

As I mentioned above, I'm well aware of the fact that this model is still very weak.

First of all, the approach of trying to quantify utility of consumption, especially of non-tangible goods might be quite inappropriate. After all, that's the major weakness of the homo economicus altogher, right?
As a defense, I'd like to see the approach chosen not as a purely numerical, but rather as a concept as whole. You can imagine and include whatever you want into this utility function.

Second, the assumptions and constraints are very restrictive. Consequently, the results might not only be inaccurate, but even wrong and misleading.
The assumptions should incrementally be loosened in further research. I intent to do so in upcoming weeks.

Third, some empirical studies should be conducted, until the theory can eventually be rejected (Karl Popper again).

I hope that I managed to make my point, and am looking forward to all kind of additional critique and feedback.


  1. Hey Simon, Savo here. In what language should I comment? :)

    I read this some time ago, but did not have some good answer... anyways it is quite a nice work. My only objection would be in kind of what you have already mentioned as first critique - I am not quite sure how can utility be quantified or measured... What do you think about posting this on: - after all it is School of Economics that holds that the complexity of subjective human choices makes mathematical modeling of the evolving market practically impossible

  2. Hi Savo,

    thanks for your suggestion, sounds like a good idea!
    Do you follow at
    I posted my question:

    Let's see what folks over there think about it!


  3. Here's what folks will think - You need to be very careful about definitions. Your definitions of the LDMU and of "good" are not correct. Furthermore, utility is not quantifiable. It is not cardinal, it is ordinal.

    I do not see what you're trying to prove. It's obvious that it makes sense to abstain from consuming an item. Don't we do it all the time?

    1. Thanks for your reply, I will re-consider in the light of what you mentioned.

      Really, is this that obvious? I had several discussions on that topic, and I found that the answer is not the clear at all.

    2. For me it seems more obvious that people are not able to abstain from anything looking at what people consum! Br ryttl

  4. I just saw it now, I hope you are satisfied with answers... and that you are not intimidated by them :D
    This was one more good remark: "Rothbard makes it clear that consuming an item today and consuming an item at some other time can very well mean that you're not consuming the same good. Like his example of "ice-in-the-winter" and "ice-in-the-summer.""

    1. Oh yes, very happy about the good feedback that helps me to improve - this was a good hint from you in the right direction!
      For Rothbard's point - yes he is right there, probably something I should either deal with or explicitly mention as hypothesis in first version.

  5. Good excercise. Math seems correct.
    The introduction of the saturation factor justifies the homo oeconomicus in a way. Any consumption behaviour can be explained by s. Got it.
    I'm wondering about the practical implications of the model? What can you actually do with this information? If you had added a paragraph on this, my saturation factor of reading your blog posts would have been lowered significantly.


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